Show commands for:
SageMath
sage: E = EllipticCurve("b1")
sage: E.isogeny_class()
Elliptic curves in class 303450.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 303450.b1 | 303450b6 | \([1, 1, 0, -12590140650, -543747958852500]\) | \(585196747116290735872321/836876053125000\) | \(315627398074255517578125000\) | \([2]\) | \(509607936\) | \(4.3560\) | |
| 303450.b2 | 303450b3 | \([1, 1, 0, -1825179650, 30007268392500]\) | \(1782900110862842086081/328139630024640\) | \(123757702521159682815000000\) | \([2]\) | \(254803968\) | \(4.0094\) | |
| 303450.b3 | 303450b4 | \([1, 1, 0, -794027650, -8334185895500]\) | \(146796951366228945601/5397929064360000\) | \(2035826332001483450625000000\) | \([2, 2]\) | \(254803968\) | \(4.0094\) | |
| 303450.b4 | 303450b2 | \([1, 1, 0, -125859650, 366029632500]\) | \(584614687782041281/184812061593600\) | \(69701779511683905600000000\) | \([2, 2]\) | \(127401984\) | \(3.6628\) | |
| 303450.b5 | 303450b1 | \([1, 1, 0, 22108350, 38872384500]\) | \(3168685387909439/3563732336640\) | \(-1344059924580925440000000\) | \([2]\) | \(63700992\) | \(3.3163\) | \(\Gamma_0(N)\)-optimal |
| 303450.b6 | 303450b5 | \([1, 1, 0, 311397350, -29720843370500]\) | \(8854313460877886399/1016927675429790600\) | \(-383533780214002738485178125000\) | \([2]\) | \(509607936\) | \(4.3560\) |
Rank
sage: E.rank()
The elliptic curves in class 303450.b have rank \(1\).
Complex multiplication
The elliptic curves in class 303450.b do not have complex multiplication.Modular form 303450.2.a.b
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
